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A theoretical approach to the interaction between buckling and resonance instabilities

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Abstract

This article deals with the interaction between buckling and resonance instabilities of mechanical systems. Taking into account the effect of geometric nonlinearity in the equations of motion through the geometric stiffness matrix, the problem is reduced to a generalized eigenproblem where both the loading multiplier and the natural frequency of the system are unknown. According to this approach, all of the forms of instabilities intermediate between those of pure buckling and pure forced resonance can be investigated. Numerous examples are analyzed, including discrete mechanical systems with one to n degrees of freedom, continuous mechanical systems, such as oscillating deflected beams subjected to a compressive axial load, as well as oscillating beams subjected to lateral–torsional buckling. A general finite element procedure is also outlined, with the possibility to apply the proposed approach to any general bi- or tri-dimensional framed structure. The proposed results provide a new insight in the interpretation of coupled phenomena such as flutter instability of long-span or high-rise structures.

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References

  1. Carpinteri A (1997) Structural mechanics: a unified approach. Chapman & Hall, London

    MATH  Google Scholar 

  2. Clough RW, Penzien J (1975) Dynamics of structures. McGraw-Hill, New York

    MATH  Google Scholar 

  3. Carpinteri A, Pugno N (2005) Towards chaos in vibrating damaged structures. Part I: Theory and period doubling. J Appl Mech 72: 511–518

    Article  MATH  Google Scholar 

  4. Carpinteri A, Pugno N (2005) Towards chaos in vibrating damaged structures. Part II: Parametrical investigation. J Appl Mech 72: 519–526

    Article  MATH  Google Scholar 

  5. Kawai H (1993) Bending and torsional vibration of tall buildings in strong wind. J Wind Eng Ind Aerodyn 50: 281–288

    Article  Google Scholar 

  6. Astiz MA (1998) Flutter stability of very long suspension bridges. ASCE J Bridge Eng 3: 132–139

    Article  Google Scholar 

  7. Scott R (2001) In the wake of Tacoma: suspension bridges and the quest for aerodynamic stability. ASCE, Reston

    Google Scholar 

  8. Scanlan RH (1996) Aerodynamics of cable-supported bridges. J Constr Steel Resour 39: 51–68

    Article  Google Scholar 

  9. Starossek U (1993) Prediction of bridge flutter through the use of finite elements. Struct Eng Rev 5: 301–307

    Google Scholar 

  10. Simiu EH, Scanlan RH (1986) Wind effects on structures. Wiley, New York

    Google Scholar 

  11. Billah KY, Scanlan RH (1991) Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks. Am J Phys 59: 118–124

    Article  ADS  Google Scholar 

  12. Scanlan RH (1994) Toward introduction of empiricism in the wind design of long-span bridges. In: Proceedings of the international conference on cable stayed and suspension bridges, pp 15–28

  13. Liapunov AM (1892) General problem of stability of motion. Kharkov (reproduced in Annals of Mathematics Studies 17, Princeton University Press, Princeton 1949)

  14. Bolotin VV (1963) Non-conservative problems of the theory of elastic stability. Pergamon Press, New York

    Google Scholar 

  15. Bolotin VV (1964) The dynamic stability of elastic systems. Holden-Day, San Francisco (Russian original: Moscow, 1956)

  16. Seyranian AP (1991) Stability and catastrophes of vibrating systems depending on parameters. Technical University of Denmark, Lyngby

    Google Scholar 

  17. Seyranian AP, Mailybaev AA (2003) Multiparameter stability theory with mechanical applications. World Scientific, Singapore

    MATH  Google Scholar 

  18. Huseyin K (1986) Multiple parameter stability theory and its applications: bifurcation, catastrophes, instabilities. Clarendon, Oxford

    Google Scholar 

  19. Huseyin K (2002) Dynamics, stability, and bifurcations. ASME Appl Mech Rev 55: R5–R15

    Article  Google Scholar 

  20. Elfelsoufi Z, Azrar L (2005) Buckling, flutter and vibration analyses of beams by integral equation formulations. Comput Struct 83: 2632–2649

    Article  Google Scholar 

  21. Bolotin VV (1995) Dynamic stability of structures. In: Kounadis AN, Kratzig WB (eds) Nonlinear stability of structures: theory and computational techniques. Springer, Wien, pp 3–72

    Google Scholar 

  22. Bathe KJ (1982) Finite element procedures in engineering analysis. Prentice-Hall, Englewood Cliff

    Google Scholar 

  23. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier, Amsterdam

    MATH  Google Scholar 

  24. Rutenberg A (1982) Simplified P-delta analysis for asymmetric structures. ASCE J Struct Div 108: 1995–2013

    Google Scholar 

  25. Przemieniecki JS (1985) Theory of matrix structural analysis. Dover, New York

    Google Scholar 

  26. Ammann OH, Von Karman T, Woodruff GB (1941) The failure of the Tacoma Narrows bridge. Report to the Federal Works Agency, March 28

  27. Scanlan RH, Tomko JJ (1971) Airfoil and bridge deck flutter derivatives. J Eng Mech 97: 1717–1737

    Google Scholar 

  28. Scanlan RH (1978) The action of flexible bridges under winds, I: flutter theory. J Sound Vib 60: 187–199

    Article  ADS  MATH  Google Scholar 

  29. Como M, Grimaldi A, Maceri F (1985) Statical behaviour of long-span cable-stayed bridges. Int J Solids Struct 21: 831–850

    Article  MATH  Google Scholar 

  30. Namini A, Albrecht P, Bosch H (1992) Finite element-based flutter analysis of cable-suspended bridges. ASCE J Struct Eng 118: 1509–1526

    Article  Google Scholar 

  31. Brancaleoni F, Diana G (1993) The aerodynamic design of the Messina Straits bridge. J Wind Eng Ind Aerodyn 48: 395–409

    Article  Google Scholar 

  32. Brancaleoni F, Brotton DM (2005) The role of time integration in suspension bridge dynamics. Int J Numer Methods Eng 20: 715–732

    Article  Google Scholar 

  33. Scanlan RH (1981) State-of-the-art methods for calculating flutter, vortex-induced and buffeting response of bridge structures. Federal Highway Administration, Report No. FHWA/RD-80/050, Washington, DC

  34. Holmes PJ (1978) Pipes supported at both ends cannot flutter. J Appl Mech 45: 619–622

    Article  MATH  Google Scholar 

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Correspondence to Marco Paggi.

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Electronic pre-print of this manuscript is available at arXiv:0802.0756v1.

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Carpinteri, A., Paggi, M. A theoretical approach to the interaction between buckling and resonance instabilities. J Eng Math 78, 19–35 (2013). https://doi.org/10.1007/s10665-011-9478-0

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  • DOI: https://doi.org/10.1007/s10665-011-9478-0

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